# Dynamics of Partial Differential Equations

## Volume 12 (2015)

### Nonnegative solutions of a fractional sub-Laplacian differential inequality on Heisenberg group

Pages: 379 – 403

DOI: http://dx.doi.org/10.4310/DPDE.2015.v12.n4.a4

#### Authors

Y. Liu (School of Mathematics and Physics, University of Science and Technology, Beijing, China)

Y. Wang (Department of Mathematics and Physics, North China Electric Power University, Beijing, China; and Department of Mathematics and Statistics, Memorial University, St. John’s, Newfoundland, Canada)

J. Xiao (Department of Mathematics and Statistics, Memorial University, St. John’s, Newfoundland, Canada)

#### Abstract

In this paper we study nonnegative solutions of\begin{align}(\dagger) & &{\lvert g \rvert}^{\gamma}_{\mathbb{H}^n} u^p \leq (- \Delta_{\mathbb{H}^n})^{\frac{\alpha}{2}} u \text{ on } \mathbb{H}^n \text{,}\end{align}where $\mathbb{H}^n$ is the Heisenberg group; ${\lvert \cdot \rvert}_{\mathbb{H}^n}$ is the homogeneous norm; $\Delta_{\mathbb{H}^n}$ is the sub-Laplacian; $(p, \alpha, \gamma) \in (1, \infty) \times (0, 2) \times [0, (p-1)Q)$; and $Q = 2n+ 2$ is the homogeneous dimension of $\mathbb{H}^n$. In particular, we prove that any nonnegative solution of $(\dagger)$ is zero if and only if $p \leq \frac{Q+\gamma}{Q-\alpha}$.

#### Keywords

Heisenberg group, nonnegative weak solution, fractional sub-Laplacian

#### 2010 Mathematics Subject Classification

Primary 35R03. Secondary 35R11.

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Published 10 December 2015